v9 update (May 2026): Solar and Cabibbo angles reframed via reciprocal amplitudes a₊²=9/20 and a₋²=1/20 (pure A₃ arithmetic). A₃ structure derived from Binary Closure (not assumed). References to Papers IV and V added. Koide proven from C (h) =0 (three lines). FRC demoted to corollary. Related papers: Paper I: 10. 5281/zenodo. 20340356 Paper III: 10. 5281/zenodo. 20284625 Paper IV: 10. 5281/zenodo. 20340084 Paper V: 10. 5281/zenodo. 20329022 We present candidate derivations of Standard Model fermion mixing parameters from the A₃ Jones subfactor within Two-Sided Closure Theory (TSCT). The paper establishes two results with explicit proof status: (i) CONDITIONAL KOIDE THEOREM. The Koide formula K = 2/3 for charged lepton masses — observed since 1982, never previously derived — follows from the A₃ Fourier Recurrence Condition, which in turn follows from Yukawa democracy in the unrendered sector (the natural absence of a preferred DFT mode before the A₃ structure is imposed) and the Pimsner–Popa exact attenuation of the oscillatory Jones-complement mode by 1/δ = 1/√2 in amplitude. The three algebraic steps: correct DFT gives |ĥ₁|/|ĥ₀| = B/2; FRC forces B = δ = √2; K = (1/3) (1 + B²/2) = 2/3. The Jones index M: N = 2 of the A₃ subfactor is directly encoded in the lepton mass spectrum. Remaining proof obligations: Yukawa democracy (O2a) and Pimsner–Popa saturation for the oscillatory mode (O2b). (ii) SOLAR ANGLE CANDIDATE. The two-sided A₃ closure formula (δ + δ⁻¹) ² / (‖d‖² ‖d⁻¹‖²) = 9/20 gives the PMNS solar mixing angle θ₁₂ = arctan √ (9/20) ≈ 33. 87° (observed: 33. 4° ± 0. 75°). The number 9/20 is rational, arising from pure A₃ Jones subfactor data — quantum dimensions (1, √2, 1), Jones index 2 — with zero free parameters. A structural lemma connecting the TSCT generation quadratic form to this two-sided product is the remaining proof obligation (O1). Four further mixing observables are discussed with explicit proof status: - Atmospheric angle θ₂₃ ≈ 45°: conditional on exact A₃ μ–τ symmetry - CP phase δCP = −π/2: structural conjecture (minimisation not constructed) - Reactor angle θ₁₃ ~ 8. 5°: open (shear coefficient not computed) - Cabibbo angle θC ≈ 12. 92°: conditional on md/mₛ = 1/20 None of these four should be regarded as derived results in the present paper. The paper is an honest map to a complete Standard Model derivation, identifying bounded proof obligations rather than claiming completion. The two genuine results — conditional Koide and the rational 9/20 solar angle candidate — are the strongest algebraic connections between the Jones subfactor structure and particle physics phenomenology currently known. This is part of the Two-Sided Closure Theory (TSCT) series. Companion paper: Einstein's Field Equations from Algebraic Closure (doi: insert Paper I DOI). Keywords: Standard Model, Koide formula, PMNS matrix, CKM matrix, Jones subfactors, A₃ subfactor, solar mixing angle, Pimsner–Popa theorem, Two-Sided Closure Theory, modular flow, Yukawa sector, neutrino mixing.
David Manton Sparks (Tue,) studied this question.